I hate to admit I don't know the answer to this but a referee has asked me about it in a paper of mine so here goes. Let F be a padic field with p odd. Let Q by a quaternary quadratic form over F of Witt rank one. Consider the orthogonal group G=O(Q,F). Then [G,G] is the kernel of the spinor norm and is isomorphic to PSL(2,E) where E is a quadratic extension of F determined by Q. My question is whether there is a nice interpretion of the spinor norm for G? For example if we replace Q by a ternary form of Witt rank one then the commutator is PSL(2,F), SO(Q,F) is PGL(2,F) and the spinor norm is given by det on PGL(2,F). Anything similar in the four dimensional case?
